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What is a mathematical proof?

A logical and convincing argument that proves that a statement is true is called a mathematical proof. When true, the proof will hold good in each and every possible case without exception. The proof is usually formulated by deductive reasoning.

What are the characteristics of a good proof?

The characteristics of a good proof are as follows:

  • Statement to be proved.
  • List of provided information.
  • The diagram to illustrate the given information.
  • Explanation of what is required to be proved.
  • Developing a method of deductive reasoning to prove the statement, such that all possible cases are covered.

Thus, a proof would be considered complete if it possesses the above parts.

What is a two - column proof?

A proof that has numbered statements on the left and reasons justifying them on the right is called a two-column proof. These statements are generally incremental in nature. That is, the result of one statement is used in the next statement. They exhibit a logical sequence of the flow of reasoning.

In short,  a two-column proof is a proof that has a number of statements on the left with corresponding reasons showing their validity on the right,  written in order, to prove that the given statement is true.

Statements

Reasons

Each statement is either the given information or the consequence of applying a known property or concept to statements already made.

Each reason is the account for the respective statement.

Note: The reasons made use of in a proof can contain definitions, theorems, properties and postulates.

What is a theorem?

A theorem is a true statement that follows as a result of other true statements. It can also be said that a theorem is a true statement supported by a proof.

What is a postulate?

A postulate is a statement that need not be proved. It is believed to be true. Most theorems are based on postulates.

What is an angle?

Consider two rays sharing the same end point. They form an angle. The common endpoint is the vertex of the angle. The rays are called the sides or arms of the angle.

Angle

How are angles measured?

Angles are measured with a protractor. The protractor has markings on the inner and outer perimeter. One ray is aligned with the base of the protractor along the ‘0’ measure of one perimeter. The reading on the same perimeter coinciding with the other ray gives the measure of the angle.

Angle

What are the different types of angles?

Angles can be classified according to their measures.

Acute Angle: An angle whose measure is less than 90º is called an acute angle.

Right Angle: An angle whose measure is exactly 90º is called a right angle.

Obtuse Angle: An angle whose measure is greater than 90º but less than 180º is called an obtuse angle.

Straight angle: An angle whose measure is exactly 180º is called a straight angle. This angle always represents a straight line.

Reflex angle: An angle whose measure is greater than 180º and less than 360º is called a reflex angle.

Total angle at a point: An angle in which both arms coincide. This angle measures 360º.

Adjacent Angles: Two angles that share a common vertex and have a side in common are called adjacent angles.

Angle

What is congruence of angles?

Two or more angles are said to be congruent when their measures are equal.

What are the properties of congruence of angles?

Properties of angle congruence are:

Reflexive Property

Symmetric Property

Transitive Property

Reflexive Property:

This property illustrates that an angle is always congruent to itself.  

Symmetric Property:

When an angle is congruent to another, then the vice versa is also true.

If A is congruent to B, then B is congruent to A.             

Transitive Property:

The transitive property holds good for congruence of angles.

A≈ B and B≈ C  then  A≈ C

 

Transitive Property of  Angle

Here is a two column proof to illustrate the transitive property of angle congruence.

Proof
Reason
angleA ≈ angleB
Given
angleB ≈ angleC  
Given
angleB ≈ angleA    
Symmetric property of angle congruence
angleC ≈ A   
Both angles are equal to B.
angleA ≈ angleC
Symmetric property of angle congruence

In proofs, one can make a single statement at a time, until the conclusion is reached.

Deductive reasoning is made use of, since statements are made based on the facts. Generally, the first statement and reason pair written is the provided information




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